Using Mathematica (Irix , version 3.0?), I have just discovered that the
theta-functions are dicontinuos, contradicting the long proven
analyticity!
You can see it by executing
ParametricPlot[
Evaluate[{Re[#], Im[#]}&@EllipticTheta[1,Pi (0.9+0.3I),-0.1+t I]],{
t,-.5,.5}]
The discontinuity occurs when t=0. (the straight segment in the plot is
another long-standing feature). Actually, a slightly more careful
analysis reveals that in Mathematica's implementation of
EllipticTheta[1,z,q], the values with Im[q]<0 are messed up. One can
repair this as follows:
ParametricPlot[
Evaluate[{Re[#], Im[#]}&@
If[t>=0,EllipticTheta[1,Pi (0.9+0.3I),-0.1+t I],
I EllipticTheta[1,Pi (0.9+0.3I),-0.1+t I]]],{t,-.5,.5}]
Enjoy!
Matthias Weber